For the \(\hat{\mathbf {s}}_{2,r}\)  in the LMMSE estimator, message passing rules have shown that it is the expectation of
\(\hat{\mathbf {s}}_{2,r} \sim \mathcal {CN}(\mathbf {s}_2;\mathbf {u}_{2,r},\gamma _{2,r}^{-1}\mathbf {I} ) \mathcal {CN}(\mathbf {v}_r;\mathbf {A}\mathbf {s}_2,{\sigma }^2\mathbf {I})\) . Herein, the pilot matrix \(\mathbf {A}\)  is updated as \(\mathbf {A}^t\triangleq \mathbf {A}\text{diag}(\beta ^t)\)  at the \(t\) th layer for a more accurate estimation by introducing the parameter \(\beta ^t=[\beta _1^t,\beta _2^t,\cdots ,\beta _N^t]\) . In other words, the parameter \(\beta _n^t\)  acts to scale the power of the \(n\) th column of pilot matrix \(\mathbf {A}\) .
Compared with the scaling rule of pilot matrix in the learned AMP (LAMP) [1]}, where all elements of matrix \(\mathbf {A}\)  vary with the layer \(t\) , the proposed scheme can decrease the expense of an \(L\) -fold in memory and training complexity.
